Abstract
A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβ η. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem.
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References
Barendregt, H.P.: The lambda calculus: Its syntax and semantics. North-Holland Publishing, Amsterdam (1984)
Bastonero, O., Gouy, X.: Strong stability and the incompleteness of stable models of λ-calculus. Annals of Pure and Applied Logic 100, 247–277 (1999)
Berardi, S., Berline, C.: βη-complete models for system F. Mathematical Structure in Computer Science 12, 823–874 (2002)
Berline, C.: From computation to foundations via functions and application: The λ-calculus and its webbed models. Theoretical Computer Science 249, 81–161 (2000)
Berline, C.: Graph models of λ-calculus at work, and variations. Math. Struct. in Comp. Science 16, 185–221 (2006)
Berline, C., Salibra, A.: Easiness in graph models. Theoretical Computer Science 354, 4–23 (2006)
Berry, G.: Stable models of typed lambda-calculi. In: Ausiello, G., Böhm, C. (eds.) Automata, Languages and Programming. LNCS, vol. 62, Springer, Heidelberg (1978)
Bucciarelli, A., Ehrhard, T.: Sequentiality and strong stability. In: Sixth Annual IEEE Symposium on Logic in Computer Science (LICS 1991), pp. 138–145. IEEE Computer Society Press, Los Alamitos (1991)
Bucciarelli, A., Salibra, A.: The minimal graph model of lambda calculus. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 300–307. Springer, Heidelberg (2003)
Bucciarelli, A., Salibra, A.: The sensible graph theories of lambda calculus. In: 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society Press, Los Alamitos (2004)
Bucciarelli, A., Salibra, A.: Graph lambda theories. Mathematical Structures in Computer Science (to appear)
Di Gianantonio, P., Honsell, F., Plotkin, G.D.: Uncountable limits and the lambda calculus. Nordic J. Comput. 2, 126–145 (1995)
Giannini, P., Longo, G.: Effectively given domains and lambda-calculus models. Information and Control 62, 36–63 (1984)
Gruchalski, A.: Computability on dI-Domains. Information and Computation 124, 7–19 (1996)
Honsell, F., Ronchi della Rocca, S.: An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus. Journal of Computer and System Sciences 45, 49–75 (1992)
Kerth, R.: Isomorphism and equational equivalence of continuous lambda models. Studia Logica 61, 403–415 (1998)
Kerth, R.: On the construction of stable models of λ-calculus. Theoretical Computer Science 269, 23–46 (2001)
Longo, G.: Set-theoretical models of λ-calculus: theories, expansions and isomorphisms. Ann. Pure Applied Logic 24, 153–188 (1983)
Plotkin, G.D.: Set-theoretical and other elementary models of the λ-calculus. Theoretical Computer Science 121, 351–409 (1993)
Salibra, A.: A continuum of theories of lambda calculus without semantics. In: 16th Annual IEEE Symposium on Logic in Computer Science (LICS 2001), pp. 334–343. IEEE Computer Society Press, Los Alamitos (2001)
Salibra, A.: Topological incompleteness and order incompleteness of the lambda calculus. ACM Transactions on Computational Logic 4, 379–401 (2003)
Scott, D.S.: Continuous lattices. In: Toposes, Algebraic geometry and Logic. LNM, vol. 274, Springer, Heidelberg (1972)
Selinger, P.: Order-incompleteness and finite lambda reduction models. Theoretical Computer Science 309, 43–63 (2003)
Stoltenberg-Hansen, V., Lindström, I., Griffor, E.R.: Mathematical theory of domains. In: Cambridge Tracts in Theoretical Computer Science, vol. 22, Cambridge University Press, Cambridge (1994)
Visser, A.: Numerations, λ-calculus and arithmetic. In: Hindley, J.R., Seldin, J.P. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pp. 259–284. Academic Press, New York (1980)
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Berline, C., Manzonetto, G., Salibra, A. (2007). Lambda Theories of Effective Lambda Models. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_22
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DOI: https://doi.org/10.1007/978-3-540-74915-8_22
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